Question

Find projvu and projuv. Use the Euclidean inner product. u = (1, −1, −2), v = (1, 1, 2)(a) projvu(b) projuv

Answer 1

Answer:

Step-by-step explanation:

Given the vectors  u = (1, −1, −2) and v = (1, 1, 2),

projection of u on v is expressed as $proj_v u = \dfrac{u*v}{|v|^2}*v$

u.v = (1, −1, −2).(1, 1, 2)

u.v = 1(1)+(-1)(1)+(-2)(2)

u.v = 1-1-4

u.v = -4

|v| = √1²+1²+ 2²

|v| = √1+1+4

|v| = √6

|v|² = (√6)²

|v|² = 6

$proj_v u = \dfrac{-4}{6}*(1,1,2)\\proj_v u = (\dfrac{-4}{6}, \dfrac{-4}{6}, \dfrac{-4}{3} )$

Similarly for projection of v on u;

$proj_u v = \dfrac{u*v}{|u|^2}*u$

|u| = √1²+(-1)²+ (-2)²

|u| = √1+1+4

|u| = √6

|u|² = (√6)²

|u|² = 6

$proj_u v = \dfrac{-4}{6}*(1,-1,-2)\\proj_u v = (\dfrac{-4}{6}, \dfrac{4}{6}, \dfrac{4}{3} )$